Derivative of Wasserstein distance $W^p_p$ along solutions of the continuity equation (contradicting statements in different sources)

Question

Let $(\rho^{(i)}_t,{\bf v}^{(i)}_t)$ for $i = 1,2$ be two solutions of the continuity equation $$\partial \rho^{(i)}_t + \nabla\cdot \left({\bf v}^{(i)}_t \rho^{(i)}_t\right) = 0 \label{1}\tag{1}$$ on a compact domain $\Omega$ (with no-flux boundary conditions), and suppose that $(\rho^{(i)}_t,{\bf v}^{(i)}_t)$ are absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}^d$ for every $t$ and that $\rho^{(i)}$ are absolutely continuous curves in $\mathbb{W}_p(\Omega)$. Then Corollary 5.25 in the monograph Optimal Transport for Applied Mathematicians stated that $$\frac{d}{dt}\left(\frac{1}{p} W^p_p\left(\rho^{(1)}_t,\rho^{(2)}_t\right)\right) = \int_{\Omega} \left(x-T_t(x)\right)\cdot \left({\bf v}^{(1)}_t(x) - {\bf v}^{(2)}_t(T_t(x))\right)\rho^{(1)}_t ~dx \label{2}\tag{2}$$ where $T_t$ is the optimal transport map from $\rho^{(1)}_t$ to $\rho^{(2)}_t$ for the cost $\frac{1}{p}|x-y|^p$.

---

On the other hand, in the paper Convergence to equilibrium in Wasserstein distance for Fokker-Planck equations the continuity equation \eqref{1} with the specific velocity field $${\bf v}_t = -(\nabla \log \rho_t + A) \label{3}\tag{3}$$ is considered. Here the vector field $A$ takes the form $A = \nabla V + F$ where $F$ satisfies $\nabla \cdot (\mathrm{e}^{-V} F) = 0$, so that the equilibrium solution/distribution is of the form $\nu = \mathrm{e}^{-V}$. Now Theorem 2.1 in the aforementioned paper claimed that $$\frac{d}{dt}\left(\frac{1}{2} W^2_2\left(\rho_t,\nu\right)\right) = -\int (x-\nabla \psi_t)\cdot \left(\nabla \log \rho_t + A\right) ~ d\rho_t\label{4}\tag{4}$$ where $\nabla \psi_t$ push forwards $\rho_t$ to $\nu$ for every $t \geq 0$.

---

Of course, we can take $T_t(x) = \nabla \psi_t(x)$ due to property of $W^2_2$. But I am failing to see the equivalence between \eqref{2} and \eqref{4} (when $p=2$). Specially, I think the formulation \eqref{2} contains the term $$\int (x-\nabla \psi_t)\cdot {\bf v}_t(T_t(x))~d\rho_t$$ while such term is missing in the formulation \eqref{4}. May I know what is going on here?

Answer 1

Without having read the sources in detail, I would notice the following:

In (4) the second measure is time independent. To satisfy (1) this then corresponds to $(\rho^{(2)}_t,v_t^{(2)}) = (\nu,0)$. But then since $v_t^{(2)}$ vanishes, so does the second term in (2).

All the other changes are already noticed in the question. One is just the specific definition of the vector field. The other is the fact that for absolutely continuous measures the unique solution $T_t$ can be expressed as a gradient $\nabla \psi_t$, where $\psi_t$ is a convex function. This is a well known theorem (due to Brenier, I believe) which is found in any textbook on optimal transport.